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Then

(i) V() 6¼ {0} iff  2 X

(ii) If  2 X

, the representation of G in V() is

equivalent to L(w

()), where w

2 W is the

unique element of the Weyl group which sends

to R



This theorem can also be reformulated in more

geometric terms: the spaces V() are naturally

interpreted as spaces of global sections of appro-

priate line bundles on the ‘‘flag variety’’

B= G

=B = G=T.

For classical groups, irreducible representations

can also be constructed explicitly as the subspaces in

tensor powers (C

)

k

, transforming in a certain way

under the action of the symmetric group S

Characters and Multiplicities

Characters

Let (, V) be a f.d. representation of G and let 



its character as defined by [5]. Since 



is central,

and every element in G is conjugate to an element of

T, 



is completely determined by its restriction to

T, which can be computed from the weight decom-

position [6]:





2XðTÞ

dim V



 e



2XðTÞ

mult



  e



½10

where e



is the function on T defined by



( exp (t)) = e

ht, i

, t 2 t. Note that e

þ

= e





and

that e

= 1.

Weyl Character Formula

Theorem 19 (Weyl character formula). Let  2 X

Then



LðÞ

þ



; A



w2W

"ðwÞe

wðÞ

where, for w 2 W, we denote "(w) = det w consid-

ered as a linear map t



, and  = (1=2)

.

In particular, computing the value of the character

at point t = 0 by L’Hopital’s rule, it is possible to

deduce the following formula for the dimension of

irreducible representations:

dim Lð Þ¼

2R

h

;þ i

h

;i

½11

Example 9 Let G = SU(2). Then Weyl character

formula gives, for irreducible representation 

with

highest weight k  !,



kþ1

 x

ðkþ1Þ

x  x

1

¼ x

þ x

k2

þþx

k

; x ¼ e

which implies dim 

= k þ 1.

Weyl character formula is equivalent to the follow-

ing formula for weight multiplicities, due to Kostant:

mult

LðÞ

 ¼

w2W

"ðwÞKðwð þ Þ  Þ

where K is Kostant’s partition function: K() is the

number of ways of writing  as a sum of positive

roots (with repetitions).

For classical Lie groups such as G = U(n), there are

more explicit combinatorial formulas for weight multi-

plicities; for U(n), the answer can be written in terms of

the number of ‘‘Young tableaux’’ of a given shape.

Details can be found in Fulton and Harris (1991).

Tensor Product Multiplicities

Let (, V) be a f.d. representation of G. By complete

reducibility, one can write V =n



L(). The coeffi-

cients n



are called multiplicities; finding them is an

important problem in many applications. In parti-

cular, a special case of this is finding the multi-

plicities in tensor product of two unirreps:

LðÞLðÞ¼





LðÞ

Characters provide a practical tool for computing

multiplicities: since characters of unirreps are line-

arly independent, multiplicities can be found from

the condition that 

=n





L()

. In particular,



LðÞ



LðÞ







LðÞ

Example 10 For G = SU(2), tensor product multi-

plicities are given by



 

¼

where the sum is taken over all l such that jm  nj

l  m þ n, m þ n þ l is even.

For G = U(n), there is an algorithm for finding the

tensor product multiplicities, formulated in the

language of Young tableaux (Littlewood–Richardson

rule). There are also tables and computer programs

for computing these multiplicities; some of them are

listed in the bibliography.

See also: Classical Groups and Homogeneous Spaces;

Combinatorics: Overview; Equivariant Cohomology and

Compact Groups and Their Representations 585

the Cartan Model; Finite Group Symmetry Breaking; Lie

Groups: General Theory; Ljusternik–Schnirelman Theory;

Noncommutative Geometry and the Standard Model;

Optimal Cloning of Quantum States; Ordinary Special

Functions; Quasiperiodic Systems; Symmetry Classes in

Random Matrix Theory.

Further Reading

Bump D (2004) Lie Groups. New York: Springer.

Bro¨ cker T and tom Dieck T (1995) Representations of Compact

Lie Groups, Graduate Texts in Mathematics, vol. 98.

New York: Springer.

Fulton W and Harris J (1991) Representation Theory. New York:

Springer.

Knapp A (2002) Lie Groups beyond an Introduction, 2nd edn.

Boston: Birkhau¨ ser.

LiE: A Computer algebra package for Lie group computations,

available from http://young.sp2mi.univ-poitiers.fr

McKay WG, Patera J, and Rand DW (1990) Tables of

Representations of Simple Lie Algebras, vol. I. Exceptional

Simple Lie Algebras. Montreal: CRM.

Serre J-P (2001) Complex Semisimple Lie Algebras. Berlin: Springer.

Simon B (1996) Representations of Finite and Compact Groups.

Providence, RI: American Mathematical Society.

Zelobenko DP (1973) Compact Lie Groups and Their Represen-

tations. Providence, RI: American Mathematical Society.

Compactification of Superstring Theory

M R Douglas, Rutgers, The State University of

New Jersey, Piscataway, NJ, USA

Introduction

Superstring theories and M-theory, at present the best

candidate quantum theories which unify gravity,

Yang–Mills fields, and matter, are directly formu-

lated in ten and eleven spacetime dimensions. To

obtain a candidate theory of our four-dimensional

universe, one must find a solution of one of

these theories whose low-energy physics is well

described by a four-dimensional effective field theory

(EFT), containing the well-established standard

model (SM) of particle physics coupled to Einstein’s

general relativity (GR). The standard paradigm for

finding such solutions is compactification, along the

lines originally proposed by Kaluza and Klein in the

context of higher-dimensional general relativity. One

postulates that the underlying D-dimensional space-

time is a product of four-dimensional Minkowski

spacetime, with a (D  4)-dimensional compact and

small Riemannian manifold K. One then finds

that low-energy physics effectively averages over K,

leading to a four-dimensional EFT whose field

content and Lagrangian are determined in terms of

the topology and geometry of K.

Of the huge body of prior work on this subject, the

part most relevant for string/M-theory is supergravity

compactification, as in the limit of low energies, small

curvatures and weak coupling, the various string

theories and M-theory reduce to ten- and eleven-

dimensional supergravity theories. Many of the quali-

tative features of string/M-theory compactification, and

a good deal of what is known quantitatively, can be

understood simply in terms of compactification of these

field theories, with the addition of a few crucial

ingredients from string/M-theory. Thus, most of this

article will restrict attention to this case, leaving many

‘‘stringy’’ topics to the articles on conformal field

theory, topological string theory, and so on. We also

largely restrict attention to compactifications based on

Ricci-flat compact spaces. There is an equally important

class in which K has positive curvature; these lead to

anti-de Sitter (AdS) spacetimes and are discussed in the

article on AdS/CFT (see AdS/CFT Correspondence).

After a general review, we begin with compacti-

fication of the heterotic string on a three complex

dimensional Calabi–Yau manifold. This was the first

construction which led convincingly to the SM, and

remains one of the most important examples. We

then survey the various families of compactifications

to higher dimensions, with an eye on the relations

between these compactifications which follow from

superstring duality. We then discuss some of the

phenomena which arise in the regimes of large

curvature and strong coupling. In the final section,

we bring these ideas together in a survey of the

various known four-dimensional constructions.

General Framework

Let us assume we are given a D-(=d þ k) dimen-

sional field theory T . A compactification is then a

D-dimensional spacetime which is topologically

the product of a d-dimensional spacetime with an

k-dimensional manifold K, the compactification or

‘‘internal’’ manifold, carrying a Riemannian metric

and with definite expectation values for all other

fields in T . These must solve the equations of motion,

and preserve d-dimensional Poincare´invariance(or,

perhaps another d-dimensional symmetry group).

586 Compactification of Superstring Theory

The most general metric ansatz for a Poincare´

invariant compactification is

f 



0 G



where the tangent space indices are 0  I < d þ

k = D,0 <d, and 1  i  k. Here 



is the

Minkowski metric, G

is a metric on K,andf is a

real-valued function on K called the ‘‘warp factor.’’

As the simplest example, consider pure

D-dimensional GR. in this case, Einstein’s equations

reduce to Ricci flatness of G

. Given our metric

ansatz, this requires f to be constant, and the metric

on K to be Ricci flat. Thus, any K which admits

such a metric, for example, the k-dimensional torus,

will lead to a compactification.

Typically, if a manifold admits a Ricci-flat metric,

it will not be unique; rather there will be a moduli

space of such metrics. Physically, one then expects

to find solutions in which the choice of Ricci-flat

metric is slowly varying in d-dimensional spacetime.

General arguments im ply that such variations

must be described by variations of d-dimensional

fields, governe d by an EFT. Given an explicit

parametrization of the family of metrics, say

(



) for some parameters 



, in principle the

EFT could be computed explicitly by promoting

the parameters to d-dimensional fields, substituting

this parametrization into the D-dimensional action,

and expanding in powers of the d-dimensional

derivatives. In pure GR, we would find the four-

dimensional effective Lagrangian

EFT

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det GðÞ

ð4Þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det GðÞ

ðÞG

ðÞ

@



@















þ ½1

While this is easily evaluated for K a symmetric space

or torus, in general a direct computation of L

EFT

impossible. This becomes especially clear when one

learns that the Ricci-flat metrics G

are not explicitly

known for the examples of interest. Nevertheless,

clever indirect methods have been found that give a

great deal of information about L

EFT

;thisismuchof

the art of superstring compactification. However, in

this section, let us ignore this point and continue as if

we could do such computations explicitly.

Given a solution, one proceeds to consider its

small perturbations, which satisfy the linearized

equations of motion. If these include exponentially

growing modes (often called ‘‘tachyons’’), the solu-

tion is unstable. (Note that this criterion is modified

for AdS compactifications). The remaining perturba-

tions can be divided into massless fields, correspond-

ing to zero modes of the linearized equations of

motion on K, and massive fields, the others. General

results on eigenvalues of Laplacians imply that the

masses of massive fields depend on the diameter of

K as m  1=diam(K), so at energies far smaller than

m, they cannot be excited (this is not universal;

given strong negative curvature on K, or a rapidly

varying warp factor, one can have perturbations of

small nonzero mass). Thus, the massive fields can be

‘‘integrated out,’’ to leave an EFT with a finite

number of fields. In the classical approximation, this

simply means solving their equations of motion in

terms of the mass less fields, and using these

solutions to eliminate them from the action. At

leading order in an expansion around a solution,

these fields are zero and this step is trivial; never-

theless, it is useful in making a systematic definition

of the interaction terms in the EFT.

As we saw in pure GR, the configuration space

parametrized by the massless fields in the EFT, is the

moduli space of compactifications obtained by

deforming the original solution. Thus, from a

mathematical point of view, low-energy EFT can

be thought of as a sort of enhancement of the

concept of moduli space, and a dictionary set up

between mathematical and physical languag es. To

give its next entry, there is a natural physical metric

on moduli space, defined by restriction from the

metric on the configuration space of the theory T ;

this becomes the sigma-model metric for the scalars

in the EFT. Because the theories T arising from

string theory are geometrically natural, this metric is

also natural from a mathematical point of view, and

one often finds that much is already known about it.

For example, the somewhat fearsome two deri vative

terms in eqn [1], are (perhaps) less so when one

realizes that this is an explicit expression for the

Weil–Petersson metric on the moduli space of Ricci-

flat metrics. In any case, knowing this dictionary is

essential for taking advantage of the literature.

Another important entry in this dictionary is that

the automorphism group of a solution translates

into the gauge gro up in the EFT. This can be either

continuous, leading to the gauge symmetry of

Maxwell and Yang–Mills theories, or discrete,

leading to discrete gauge symmetry. For example, if

the metric on K has continuous isometry group G,

the resulting EFT will have gauge symmetry G,asin

the original example of Kaluza and Klein with K ﬃ S

and G ﬃ U(1). Ma thematically, these phenomena

of ‘‘enhanced symmetry’’ are often treated using the

languages of equivariant theories (cohomology,

K-theory, etc.), stacks, and so on.

Compactification of Superstring Theory 587

To give another example, obstructed deformations

(solutions of the linearized equations which do not

correspond to elements of the tangent space of the

true moduli space) correspond to scalar fields which,

while massless, appear in the effective potential in a

way which prevents giving them expectation values.

Since the quadratic terms V

are masses, this

dependence must be at cubic or higher order.

While the preceding concepts are general and apply

to compactification of all local field theories, string

and M-theory add some particular ingredients to this

general recipe. In the limits of small curvatures and

weak coupling, string and M-theory are well described

by the ten- and 11-dimensional supergravity theories,

and thus the string/M-theory discussion usually starts

with Kaluza–Klein compactification of these theories,

which we denote I, IIa, IIb, HE, HO and M. Let us

now discuss a particular example.

Calabi–Yau Compactification

of the Heterotic String

Contact with the SM requires finding compactifications

to d = 4 either without supersymmetry, or with at most

N = 1 supersymmetry, because the SM includes chiral

fermions, which are incompatible with N > 1. Let us

start with the E

 E

heterotic string or ‘ ‘HE’’ theory.

This choice is made rather than HO because only in this

case can we find the SM fermion representations as

subrepresentations of the adjoint of the gauge group.

Besides the metric, the other bosonic fields of the HE

supergravity theory are a scalar  called the dilaton,

Yang–Mills gauge potentials for the group G  E



, and a 2-form gauge potential B (often called the

‘‘Neveu–Schwarz’’ or ‘‘NS’’ 2-form) whose defining

characteristic is that it minimally couples to the

heterotic string world-sheet. We will need their gauge

field strengths below: for Yang–Mills, this is a 2-form

with a indexing the adjoint of Lie G, and for the NS

2-form this is a 3-form H

IJK

. Denoting the two

Majorana–Weyl spinor representations of SO(1, 9) as

S and C, then the fermions are the gravitino

S  V, a spin 1/2 ‘‘dilatino’’  2 C, and the adjoint

gauginos 

2 S.Weuse

to denote Dirac matrices

contracted with a ‘‘zehnbein,’’ satisfying {

, 

} =

,and

= (1=2)[

, 

], etc.

A local supersymmetry transformation with para-

meter  is then



¼D

 þ

IJK



 ½2

 ¼@



 

IJK



IJK

 ½3



¼ F



 ½4

We now assume N = 1 supers ymmetry. An unbroken

supersymmetry is a spinor  for which the left-hand

side is zero, so we seek compactifications with a

unique solution of these equatio ns.

We first discuss the case H = 0. Setting 



eqn [2] to zero, we find that the warp factor f must

be constant. The vanishing of 

requires  to be a

covariantly constant spinor. For a six-dimensional

M to have a unique such spinor, it must have SU(3)

holonomy; in other words, M must be a Calabi–Yau

manifold. In the following, we use basic facts about

their geometry.

The vanishing of  then requires constant dilaton

, while the vanishing of 

requires the gauge field

strength F to solve the hermitian Yang–Mills

equations,

2;0

¼ F

0;2

¼ F

1;1

¼ 0

By the theorem of Donaldson and Uhlenbeck–Yau,

such solutions are in one-to-one correspondence

with -stable holomorphic vector bundles with

structure group H contained in the complexification

of G. Choose such a bundle E; by the general

discussion above, the commutant of H in G will be

the automorphism group of the connection on E and

thus the low-energy gauge group of the resulting

EFT. For example, since E

has a maximal E



SU(3) subgroup, if E has structure group H = SL(3),

there is an embedding such that the unbroken gauge

symmetry is E

 E

, realizing one of the standard

grand unified groups E

as a factor.

The choice of E is constrained by anomaly

cancellation. This discussion (Green et al. 1987)

modifies the Bianchi identity for H to

dH ¼ tr R ^ R 

^ F

½5

where R is the matrix of curvature 2-forms. The

normalization of the F ^ F term is such that if we

take E ﬃ TK the holomorphic tangent bundle of K,

with isomorphic connection, then using the embed-

ding we just discussed, we obtain a solution of eqn

[5] with H = 0.

Thus, we have a complete solution of the

equations of motion. General arguments imply that

supersymmetric Minkowski solutions are stable, so

the small fluctuations consist of massless and

massive fields. Let us now discuss a few of the

massless fields. Since the EFT has N = 1 super-

symmetry, the massless scalars live in chiral multi-

plets, which are local coordinates on a complex

Ka¨ hler manifold.

First, the moduli of Ricci-flat metrics on K will

lead to massless scalar fields: the complex structure

588 Compactification of Superstring Theory

moduli, which are naturally complex, and Ka¨hler

moduli, which are not. However, in string compac-

tification the latter are complexified to the periods of

the 2-form B þ iJ integrated over a basis of H

(K, Z),

where J is the Ka¨hler form and B is the NS 2-form. In

addition, there is a complex field pairing the dilaton

(actually, exp()) and the ‘‘model-independent

axion,’’ the scalar dual in d = 4 to the 2-form B



Finally, each complex modulus of the holomorphic

bundle E will lead to a chiral multiplet. Thus, the

total number of massless uncharged chiral multiplets

is 1 þ h

1, 1

(K) þ h

2, 1

(K) þ dim H

(K,End (E)).

Massless charged matter will arise from zero

modes of the gauge field and its supersymmetric

partner spinor 

. It is slightly easier to discuss the

spinor, and then appeal to supersymmetry to get the

bosons. Decomposing the spinors of SO(6) under

SU(3), one obtains (0, p) forms, and the Dirac

equation becomes the condition that these forms

are harmonic. By the Hodge theorem, these are in

one-to-one correspondence with classes in Dolbeault

cohomology H

0, p

(K, V), for some bundle V. The

bundle V is obtained by decomposing the spinor into

representations of the holonomy group of E. For

H = SU(3), the decomposition of the adjoint under

the embedding of SU(3)  E

in E

248 ¼ð8; 1Þþð1; 78Þþð3; 27Þþð



27Þ½6

implies that charged matter will form ‘‘generations’’

in the 27, of number dim H

0, 1

(K, E), and ‘‘antigene-

rations’’ in the



27, of number dim H

0, 1

(K,



E) =

dim H

0, 2

(K, E ). The difference in these numbers is

determined by the Atiyah–Singer index theorem to be

gen

 N

 N



ðEÞ

In the special case of E ﬃ TK, these numbers are

separately determined to be N

= b

1, 1

and



= b

2, 1

, so their difference is (K)=2, half the

Euler number of K. In the real world, this number is

gen

= 3, and matching this under our assumptions

so far is very constraining.

Substituting these zero modes into the ten-

dimensional Yang–Mills action and integrating, one

can derive the d = 4 EFT. For example, the cubic

terms in the superpotential, usually called Yukawa

couplings after the corresponding fermion–boson

interactions in the component Lagrangian, are

obtained from the cubic product of zero modes

 ^ tr 

^ 

ðÞ

where  is the holomorphic 

2 H

0, 1

(K, Rep E) are

the zero modes, and tr arises from decomposing the

cubic group invariant.

Note the very important fact that this expression

only depends on the cohomology classes of the 

(and ). This means the Yukawa couplings can be

computed without finding the explicit ha rmonic

representatives, which is not possible (we do not

even know the explicit metric). More generally, one

expects to be able to explicitly compute the super-

potential and all other holomorphic quantities in

the effective Lagrangian solely from ‘‘topological’’

information (the Dolbeault cohomology ring, and

its generalizations within topological string theory).

On the other hand, computing the Ka¨ hler metric in

an N = 1 EFT is usually out of reach as it would

require having explicit normalized zero modes.

Most results for this metric come from considering

closely related compactifications with extended

supersymmetry, and argui ng that the breaking

to N = 1 supersymmetry makes small corrections

to this.

There are several generalizations of this construc-

tion. First, the necessary condition to solve eqn [5] is

that the left-hand side be exact, which requires

ðEÞ¼c

ðTKÞ½7

This allows for a wide variety of E’s to be used, so

that N

gen

= 3 can be attained with many more K’s.

This class of models is often called ‘‘(0, 2) compacti-

fication’’ to denote the world-sheet supersymmetry

of the heterotic string in these backgrounds. One can

also use bundles with larger structure group; for

example, H = SL(4) leads to unbroken SO(10)  E

and H = SL(5) leads to unbroken SU(5)  E

The subsequent breaking of the grand unified

group to the SM gauge group is typically done by

choosing K with nontrivial 

, so that it admits a

flat line bundle W with nontrivial holonomy

(usually called a ‘‘Wilson line’’). One then uses the

bundle E  W in the above discussion, to obtain the

commutant of H  W as gauge group. For example,

if 

(K) ﬃ Z

, one can use W whose holonomy is an

element of order 5 in SU(5), to obtain as commutant

the SM gauge group SU(3)  SU(2)  U(1).

Another generalization is to take the 3-form H 6¼ 0.

This discussion begins by noting that, for super-

symmetry, we still require the existence of a unique

spinor ; however, it will no longer be covariantly

constant in the Levi-Civita connection. One way to

structure the problem is to note that the right-hand

side of eqn [2] takes the form of a connection with

torsion; the resulting equations have been discussed

mathematically in (Li and Yau 2004).

Another recent approach to these compactifica-

tions (Gauntlett 2004) starts out by arguin g that 

cannot vanish on K, so it defines a weak SU(3)

structure, a local reduction of the structure group of

Compactification of Superstring Theory 589

TKto SU(3) which need not be integrable. This

structure must be present in all N = 1, d = 4 super-

symmetric compactifications and there are hopes

that it will lead to a useful classification of the

possible local structures and corresponding partial

differential equations (PDEs) on K.

Higher-Dimensional and Extended

Supersymmetric Compactifications

While there are similar quasirealistic constructions

which start from the other string theories and

M-theory, before we discuss these, let us give an

overview of compactifications with N  2 super-

symmetry in four dimensions, and in higher dimen-

sions. These are simpler analog models which can be

understood in more depth; their study led to one of

the most important discoveries in string/M-theory,

the theory of superstring duality.

As before, we require a covariantly constant

spinor. For Ricci-flat K with other background

fields zero, this requires the holonomy of K to be

one of trivial, SU(n), Sp(n), or the exceptional

holonomies G

or Spin(7). In Table 1 we tabulate

the possibilities with spacetime dimension d greater

or equal to 3, listing the supergravity theory, the

holonomy type of K, and the type of the resulting

EFT: dimension d, total number of real super-

symmetry parameters Ns, and the number of spinor

supercharges N (in d = 6, since left- and right-

chirality Majorana spinors are inequivalent, there

are two numbers).

The structure of the resulting supergravity EFTs is

heavily constrained by Ns. We now discuss the

various possibilities.

Ns = 32

Given the supersymmetry algebra, if such a super-

gravity exists, it is unique. Thus, toroidal compac-

tifications of d = 11 supergravity, IIa and IIb

supergravity lead t o the same series of maximally

supersymmetric theories. Their structure is g ov-

erned by the exceptional Lie algebra E

11d

;the

gauge charges transform in a fundamental repre-

sentation of this algebra, while the scalar fields

parametrize a coset space G=H,whereG is the

maximally split real form of the Lie group E

11d

and H is a maximal compact subgroup of G.

Nonperturbative duality symmetries lead to a

further identification by a maximal discrete sub-

group of G.

Ns = 16

This supergravity can be coupled to maximally

supersymmetric super Yang–Mills theory, which

given a choice of gauge group G is unique. Thus,

these theories (with zero cosmological constant and

thus allowing super-Poincare´ symmetry) are

uniquely determined by the choice of G.

In d = 10, the choices E

 E

and Spin(32)=Z

which arise in string theory, are almost uniquely

determined by the Green–Schwarz anomaly cancel-

lation analysis. Compactification of these HE, HO

and type I theories on T

produces a unique theory

with moduli space

 SOðn ; n þ 16; ZÞnSOðn; n þ 16; RÞ=SOðn; RÞ

 SOð n þ 16; RÞ½8

In Kaluza–Klein (KK) reduction, this arises from the

choice of metric g

, the antisymmetric tensor B

and

the choice of a flat E

 E

or Spin(32)=Z

connec-

tion on T

, while a more unified description follows

from the heterotic string world-sheet analysis. Here

the group SO(n, n þ 16) is defined to preserve an even

self-dual quadratic form  of signature (n, n þ 16);

for example,  = (E

)  (E

)  I  I  I,whereI

is the form of sig nature (1,1) and E

is the Cartan

matrix. In f act, all such forms are equivalent under

orthogonal integer similarity transform ation; so,

the resulting EFT is unique. It has a rank 16 þ 2n

gauge group, which at generic points in moduli

space is U(1)

16þ2n

, but is enhanced to a nonabelian

group G at special points. To describe G,wefirst

note that a point p in moduli s pace determines an

n-dimensional subspace V

of R

16þ2n

,and

an orthogonal subspace V

(of varying dimen-

sion). Lattice points of length squared 2 con-

tained in V

then correspond to roots of the Lie

algebra of G

Table 1 String/M-theories, holonomy groups and the resulting

supersymmetry

Theory Holonomy d Ns N

M, II Torus Any 32 Max

M SU(2) 7 16 1

SU(3) 5 8 1

441

Sp(4) 3 6 3

SU(4) 3 4 2

Spin(7) 3 2 1

IIa SU(2) 6 16 (1, 1)

SU(3) 4 8 2

342

IIb SU(2) 6 16 (0, 2)

SU(3) 4 8 2

342

HE, HO, I Torus Any 16 Max/2

SU(2) 6 8 1

SU(3) 4 4 1

321

590 Compactification of Superstring Theory

The other compactifications with Ns = 16 is

M-theory on K3 and its further toroidal reductions,

and IIb on K3. M-theory compactification to d = 7

is dual to heterotic on T

, with the same moduli

space and enhanced gauge symmetry. As we discuss

at the end of the section ‘‘String y and quantum

corre ctions, ’’ the extra mass less gauge bos ons of

enhanced gauge symmetry are M2 branes wrapped

on 2-cycles with topology S

. For such a cycle to

have zero volume, the integral of the Ka¨ hler form

and holomorphic 2-form over the cycle must vanish;

expressing this in a basis for H

(K3, R) leads to

exactly the same condition we discussed for

enhanced gauge symmetry above. The final result is

that all such K3 degenerations lead to one- of the

two-dimensional canonical singulariti es, of types A,

D or E, and the corresponding EFT phenomenon is

the enhanced gauge symmetry of corresponding

Dynkin type A, D, or E.

IIb on K3 is similar, but reducing the self-dual

Ramond–Ramond (RR) 4-form potent ial on the 2-

cycles leads to self-dual tensor multi plets instead of

Maxwell theory. The moduli space is eqn [8] but

with n = 5, not n = 4, incorporating periods of RR

potentials and the SL(2, Z) duality symmetry of IIb

theory.

One may ask if the Ns = 16 I/HE/HO theories in

d = 8 and d = 9 have similar duals. For d = 8, these

are obtained by a pretty construction known as

‘‘F-theory.’’ Geometrically, the simplest definition of

F-theory is to consider the speci al case of M-theory

on an elliptically fibered Calabi–Yau, in the limit

that the Ka¨hler modulus of the fiber becomes small.

One check of this claim for d = 8 is that the moduli

space of elliptically fibered K3s agrees with eqn [8]

with n = 2.

Another definition of F-theory is the particular

case of IIb compactification using Dirichlet

7-branes, and orientifold 7-planes. This construction

is T-dual to the type I theory on T

, which provides

its simplest string theory definition. As discussed in

Polchinski (1999), one can think of the open strings

giving rise to type I gauge symmetry as living on 32

Dirichlet 9-branes (or D9-branes) and an orientifold

nineplane. T-duality converts Dirichlet and orienti-

fold p-branes to (p  1)-branes; thus this relation

follows by applying two T-dualities.

These compactifications can also be parametrized

by elliptically fibered Calabi–Yaus, where K is the

base, and the branes correspond to singularities of

the fibration. The relation between these two

definitions follows fairly simply from the duality

between M-theory on T

, and IIb string on S

. There

is a partially understood generalization of this

to d = 9.

Finally, these constructions admit further discrete

choices, which break some of the gauge symmetry.

The simplest to explain is in the toroidal compacti-

fication of I/HE/HO. The moduli space of theories

we discussed uses flat connections on the torus

which are continuously connected to the trivial

connection, but in general the moduli space of flat

connections has other components. The simplest

example is the moduli space of flat E

 E

connections on S

, whi ch has a second component

in which the holonomy exchanges the two E

’s. On

, there are connections for which the holonomies

cannot be simultaneously diagonalized. This struc-

ture and the M-theory dual of these choices is

discussed in (de Boer et al. 2001).

Ns = 8, d < 6

Again, the gravity multiplet is uniquely determined,

so the most basic classification is by the gauge group

G. The full low-energy EFT is determined by the

matter content and action, and there are two types

of matter multiplets. First, vector multiplets contain

the Yang–Mills fields, fermions and 6  d scalars;

their action is determined by a prepotential which is

a G-invariant function of the fields. Since the vector

multiplets contain massless adjoint scalars, a generic

vacuum in which these take nonzero distinct

vacuum expectation values (VEVs) will have U(1)

gauge symmetry, the commutant of G with a generic

matrix (for d < 5, while there are several real

scalars, the potential forces these to commute in a

supersymmetric vacuu m). Vacua with this type of

gauge symmetry breaking, which does not reduce

the rank of the gauge group, are usually referred to

as on a ‘‘Coulomb branch’’ of the moduli space. To

summarize, this sector can be specified by n

, the

number of vector multiplets, and the prepote ntial F,

a funct ion of the n

VEVs which is cubic in d = 5,

and holomorphic in d = 4.

Hypermultiplets contain scalars which parame-

trize a quaternionic Ka¨ hler manifold, and partner

fermions. Thus, this sector is specified by a 4n

real

dimensional quaternionic Ka¨ hler mani fold. The G

action comes with triholomorphic moment maps; if

nontrivial, VEVs in this sector can break gauge

symmetry and reduce it in rank. Such vacua are

usually referred to as on a ‘‘Higgs branch.’’

The basic example of these compactifications is

M-theory on a Calabi–Yau 3-fold (CY

). Reduction

of the 3-form leads to h

1, 1

(K) vector multiplets,

whose scalar components are the CY Ka¨hler moduli.

The CY complex structure moduli pair with periods

of the 3-form to produce h

2, 1

(K) hypermultiplets.

Enhanced gauge symmetry then appears when the

Compactification of Superstring Theory 591

contains ADE singularities fibered over a curve,

from the same mechanism involving wrapped M2

branes we discussed under Ns = 16. If degenerating

curves lead to other singularities (e.g., the ODP or

‘‘conifold’’), it is possible to obtain extremal transi-

tions which translate physically into Coulomb–Higgs

transitions. Finally, singularities in which surfaces

degenerate lead to nontrivial fixed-point theories.

Reduction on S

leads to IIa on CY

, with the

spectrum abo ve plus a ‘‘universal hypermultiplet’’

which includes the dilaton. Perhaps the most

interesting new feature is the presence of world-

sheet instantons, which correct the metric on vector

multiplet moduli space. This metric satisfies the

restrictions of special geometry and thus can be

derived from a prepotential.

The same theory can be obtained by compactifi-

cation of IIb theory on the mirror CY

. Now vector

multiplets are related to the complex structure

moduli space, while hypermultiplets are related to

Ka¨ hler moduli space. In this case, the prepotential

derived from variation of complex structure receives

no instanton corrections, as we discuss in the next

section.

Finally, one can compactify the heterotic string on

K3  T

6d

, but this theory follows from toroidal

reduction of the d = 6 case we discuss next.

Ns = 8, d = 6

These supergravities are similar to d < 6, but there

is a new type of matter multi plet, the self-dual

tensor (in d < 6 this is dual to a vector multiplet).

Since fermions in d = 6 are chiral, there is an

anomaly cancellation condition relating the numbers

of the three types of multiplets (Aspinwall 1996,

section 6.6),

 n

þ 29n

¼ 273 ½9

One class of examples is the heterotic string

compactified on K3. In the original perturbative

constructions, to satisfy eqn [7], we need to choose a

vector bundle with c

(V) = (K3) = 24. The result-

ing degrees of freedom are a single self-dua l tensor

multiplet and a rank-16 gauge group. More gen-

erally, one can introduce N

heterotic 5-branes,

which generalize eqn [7] to c

(E) þ N

= c

(TK).

Since this brane carries a self-dual tensor multiplet,

this series of models is parametrized by n

. They are

connected by transitions in which an E

instanton

shrinks to zero size and becomes a 5-brane; the

resulting decrease in the dimension of the moduli

space of E

bundles on K3 agrees with eqn [9].

Another class of examples is F-theory on an

elliptically fibered CY

. These are related to

M-theory on an elliptically fibered CY

in the same

general way we discussed under Ns = 16. The

relation between F-theory and the heterotic string

on K3 can be seen by lifting M-theory-heterotic

duality; this suggests that the two constructions are

dual only if the CY

is a K3 fibration as well. Since

not all elliptically fibered CY

s are K3 fibered, the

F-theory construction is more general.

We return to d = 4 and Ns = 4 in the final section.

The cases of Ns < 4 which exist in d  3 are far less

studied.

Stringy and Quantum Corrections

The D-dimensional low-energy effect ive supergrav-

ity actions on which we based our discussion so far

are only approximations to the general story of

string/M-theory compactification. However, if

Planck’s constant is small, K is sufficiently large,

and its curvature is small, then they are controlled

approximations.

In M-theory, as in any theory of quantum gravity,

corrections are controlled by the Planck scale

parameter M

D2

, which sits in front of the Einstein

term of the D-dimensional effective Lagrangian, and

plays the role of h. In general, this is different from

the four-dimensional Planck scale, which satisfies

= Vol(K) M

D2

. After taking the low-energy

limit E  M

, the remaining corrections are con-

trolled by the dimensionless parameters l

=R, where

R can any characteristic length scale of the solution:

a curvature radius, the length of a nontrivial cycle,

and so on.

In string theory, one usually thinks of the

corrections as a double series expansion in g

, the

dimensionless (closed) string coupling constant, and



, the inverse string tension parameter, of dimen-

sions (length)

. The ten-dimensional Planck scale is

related to these parameters as M

= 1=g

(

)

,upto

a constant factor that depends on conventions.

Besides perturbative corrections, which have power-

like dependence on these parameters, there can be

world sheet and ‘‘brane’’ instanton corrections. For

example, a string world sheet can wrap around a

topologically nontrivial spacelike 2-cycle  in K,

leading to an instanton correction to the effective

action which is suppressed as exp(Vol()=2

More generally, any p-brane wrapping a p-cycle

can produce a similar effect. As for which terms in

the effective Lagrangian receive corrections, this

depends largely on the number and symmetries of

the fermion zero modes on the instanton world

volumes.

Let us start by discussing some cases in which one

can argue that these corrections are not present.

592 Compactification of Superstring Theory

First, extended supersymmetry can serve to elim-

inate many corrections. This is analogous to the

familiar result that the superpotential in d = 4, N = 1

supersymmetric field theory does not receive (or ‘‘is

protected from’’) perturbative corrections, and in

many cases follows from similar formal arguments.

In particul ar, supersymmetry forbids corrections to

the potential and two derivative terms in the

Ns = 32 and Ns = 16 Lagrangians.

In Ns = 8, the superpotential is protected, but the

two derivative terms can receive corrections. How-

ever, there is a simple argument which precludes

many such corrections – sinc e vector multiplet and

hypermultiplet moduli spaces are decoupled, a

correction whose control parameter sits in (say) a

vector multiplet, cannot affect hypermultiplet mod-

uli space. This fact allows for many exact computa-

tions in these theories.

As an example, in IIb on CY

, the metric on

vector multiplet moduli space is precisely eqn [1] as

obtained from supergravity (in other words, the

Weil–Petersson metric on complex structure moduli

space). First, while in principle it could have been

corrected by world-sheet instantons, since these

depend on Ka¨hler moduli which sit in hypermulti-

plets, it is not. The only other instantons with the

requisite zero modes to mod ify this metric are

wrapped Dirichlet branes. Since in IIb theory these

wrap even-dimensional cycles, they also depend on

Ka¨ hler moduli and thus leave vector moduli space

unaffected.

As previously discussed, for K3-fibered CY

, this

theory is dual to the heterotic strin g on K3  T

There, the vector multiplets arise from Wilson lines

on T

, and reduce to an adjoint multiplet of N = 2

supersymmetric Yang–Mills theory . Of course, in

the quantum theory, the metri c on this moduli space

receives instanton corrections. Thus, the duality

allows deriving the exact moduli space metric, and

many other results of the Seiberg–Witten theory of

N = 2 gauge theory, as aspects of the geometry of

Calabi–Yau moduli space.

In Ns = 4, only the superpotential is protected,

and that only in perturbation theory; it can receive

nonperturbative corrections. Indeed, it appears that

this is fairly generic, suggesting that the effective

potentials in these theories are often sufficiently

complicated to exhibit the structure required for

supersymmetry breaking and the other symmetry

breakings of the SM. Under standing this is an active

subject of research.

We now turn from corrections to novel physical

phenomena which arise in these regimes. While this

is too large a sub ject to survey here, one of the basic

principles which governs this subject is the idea that

string/M-theory compactification on a singular

manifold K is typically consistent, but has new

light degrees of freedom in the EFT, not predicted

by KK arguments. We implicitly touched on one

example of this in the discussion of M-theory

compactification on K3 above, as the space of

Ricci-flat K3 metrics has degeneration limits in

which curvatures grow without bound, while the

volumes of 2-cycles vanish. On the other hand, the

structure of Ns = 16 supersymmetry essentially

forces the d = 7 EFT in these limits to be non-

singular. Its only noteworthy feature is that a

nonabelian gauge symmetry is restored, and thus

certain charged vector bosons and their superpart-

ners become massless.

To see what is happening microscopically, we

must consider an M-theory membrane (or 2-brane),

wrapped on a degenerating 2-cycle. This appears as

a particle in d = 7, charged under the vector

potential obtained by reduct ion of the D = 11

3-form potential. The mass of this particle is the

volume of the 2-cycle multiplied by the membrane

tension, so as this volume shrinks to zero, the

particle becomes massless. Thus, the physics is also

well defi ned in 11 dimensions, though not literally

described by 11-dimensional supergravity.

This phenomenon has numerous generalizations.

Their common point is that, since the essential

physics involves new light degrees of freedom, they

can be understood in terms of a lower-dimensional

quantum theory associated with the region around

the singularity. Depending on the geometry of the

singularity, this is sometimes a weakly coupled field

theory, and sometimes a nontrivial conformal field

theory. Occasionally, as in IIb on K3, the lightest

wrapped brane is a string, leading to a ‘‘little string

theory’’ (Aharony 2000).

N = 1 Supersymmetry in Four Dimensions

Having described the general framework, we con-

clude by discussing the various constructions which

lead to N = 1 supersymmetry. Besides the heterotic

string on a CY

, these compactifications include

type IIa and IIb on orientifolds of CY

, the related

F-theory on elliptically fibered Calab i–Yau 4-folds

ðCY

Þ, and M-theory on G

manifolds. Let us briefly

spell out their ingredients, the known nonperturbative

corrections to the superpotential, and the duality

relations between these constructions.

To start, we recap the heterotic string construc-

tion. We must specify a CY

K, and a bundle E over

K which admits a Hermitian Yang–Mills connec-

tion. The gauge group G is the commutant of the

structure group of E in E

 E

or Spin(32)=Z

Compactification of Superstring Theory 593

while the chiral matter consists of metric moduli of

K, and fields corresponding to a basis for the

Dolbeault coho mology group H

0, 1

(K, Rep E) where

Rep E is the bundle E embedded into an E

bundle

and decomposed into G-reps.

There is a general (though somewhat formal)

expression for the superpotential,

W ¼

 ^þtr



A þ





 ^ H

ð3Þ

þ W

½10

The first term is the holomorphic Chern–Simons

action, whose variation enforces the F

0, 2

= 0 condi-

tion. The second is the ‘‘flux superpotential,’’ while

the third term is the nonperturbative corrections.

The best understood of these arise from super-

symmetric gauge theory sectors. In some, but not all,

cases, these can be unde rstood as arising from gauge

theoretic instantons , which can be shown to be dual

to heterotic 5-branes wrapped on K. Heterotic

world-sheet instantons can also contribute.

The HO theory is S-dual to the type I string, with

the same gauge group, realized by open strings on

Dirichlet 9-branes. This construction involves essen-

tially the same data. The two classes of heterotic

instantons are dual to D1- and D5-brane instantons,

whose world-sheet theori es are somewhat simpler.

If the CY

K has a fibration by tori, by applying

T-duality to the fibers along the lines discussed for

tori under Ns = 16 above, one obtains various type II

orientifold compactifications. On an elliptic fibra-

tion, double T-duality produces a IIb compactifica-

tion with D7s and O7s. Using the relation between

IIb theory on T

and F-theory on K3 fiberwise, one

can also think of this as an F-theory compactifica-

tion on a K3-fibered CY

. More generally, one

can compactify F theory on any elliptically fibered

4-fold to obtain N = 1. These theories have

D3-instantons, the T-duals of both the type I

D1- and D5-brane instantons.

The theory of mirror symmetry predicts that all

s have T

fibration structures. Applying the

corresponding triple T-duality, one obtains a IIa

compactification on the mirror CY

K, with D6-

branes and O6-planes. Supersymmetry requires

these to wrap special Lagrangian cycles in

K.Asin

all Dirichlet brane constructions, enhanced gauge

symmetry arises from coincident branes wrapping

the same cycle, and only the classical groups are

visible in perturbation theory. Exceptional gauge

symmetry arises as a strong coupling phenomenon

of the sort described in the previous section. The

superpotential can also be thought of as mirror to

eqn [10], but now the first term is the sum of a real

Chern–Simons action on the special Lagrangian

cycles, with disk world-sheet instanton corrections,

as studied in open string mirror symmetry. The

gauge theory instantons are now D2-branes.

Using the duality relation between the IIa string and

11-dimensional M-theory, this construction can be

lifted to a compactification of M-theory on a seven-

dimensional manifold L, which is an S

fibration over

K. The D6 and O6 planes arise from singularities in the

fibration. Generically, L can be smooth, and the

only candidate in Table 1 for such an N = 1

compactification is a manifold with G

holonomy;

therefore, L must have such holonomy. Finally, both

the IIa world-sheet instantons and the D2-brane

instantons lift to membrane instantons in M-theory.

This construction implicitly demonstrates the exis-

tence of a large number of G

holonomy manifolds.

Another way to arrive at these is to go back to the

heterotic string on K, and apply the duality (again

under Ns = 16) between heterotic on T

and M-theory

on K3 to the T

fibration structure on K,toarriveat

M-theory on a K3-fibered manifold of G

holonomy.

Wrapping membranes on 2-cycles in these fibers, we

can see enhanced gauge symmetry in this picture fairly

directly.Itisanilluminatingexercisetoworkthrough

its dual realizations in all of these constructions.

Our final construction uses the interpretation of the

strong coupling limit of the HE theory as M-theory on

a one-dimensional interval I, in which the two E

factors live on the two boundaries. Thus, our original

starting point can also be interpreted as the heterotic

string on K  I. This construction is believed to be

important physically as it allows generalizing a

heterotic string tree-level relation between the gauge

and gravitational couplings which is phenomenologi-

cally disfavored. One can relate it to a IIa orientifold as

well, now with D8- and O8-branes.

These multiple relations are often referred to as the

‘‘web’’ of dualities. They lead to numerous relations

between compactification manifolds, moduli spaces,

superpotentials, and other properties of the EFTs,

whose full power has only begun to be appreciated.

Suggestions for further reading

Original references for all but the most recent of

these topics can be found in the following textbooks

and proceedings. We have also referenced a few

research articles which are good starting points for

the more recent literature. There are far more

reviews than we could reference here, and a partial

listing of these appears at http://www.slac.stanford.

edu/spires/reviews/

See also: Brane Construction of Gauge Theories;

Random Algebraic Geometry, Attractors and Flux Vacua;

594 Compactification of Superstring Theory